Nintendo Switch battery almost dead, TV mode still possible?

Got a Nintendo Switch where the battery isn't the best anymore. In other words: A full battery lasts max. 45 minutes (playing time) Normally it was around 3-5 hours at the beginning (depending on the game)

Now my question: if at some point the battery completely gives up, can I still continue playing in TV mode (switch in the docking station / power supply). Or does the switch battery have to be replaced?

Should you actually be able to, it would also be good if the battery was also removed, it could leak. Or simply changing the battery, it's not complicated.

Change the battery